Lattice permutations and Poisson-Dirichlet distribution of cycle lengths

TL;DR

This paper investigates the distribution of cycle lengths in spatial permutations on Z^3 with a penalty factor, revealing a phase transition and Poisson-Dirichlet distribution of long cycles supported by heuristic and numerical evidence.

Contribution

It demonstrates that long cycle lengths follow a Poisson-Dirichlet distribution in a spatial permutation model with a phase transition, supported by heuristic arguments and numerical data.

Findings

Macroscopic cycles emerge below a critical temperature T.

Cycle lengths follow a Poisson-Dirichlet distribution.

Heuristic coagulation-fragmentation process explains the distribution.

Abstract

We study random spatial permutations on Z^3 where each jump x -> \pi(x) is penalized by a factor exp(-T ||x-\pi(x)||^2). The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.